Problem Analysis

To find the number of integers that are divisible by a set of numbers {2, 3, 4, 5, 6}, we first need to find the Least Common Multiple (LCM) of these numbers. Any number divisible by all of them must also be divisible by their LCM.

1. Calculating the Least Common Multiple (LCM)

We need to find the LCM of 2, 3, 4, 5, and 6.

• Prime factorization of 2: 2
• Prime factorization of 3: 3
• Prime factorization of 4: 2²
• Prime factorization of 5: 5
• Prime factorization of 6: 2 × 3

The LCM is the product of the highest power of each prime factor present in the numbers.

LCM = 2² × 3¹ × 5¹ = 4 × 3 × 5 = 60.

So, we are looking for the number of multiples of 60 between 1 and 1000.

2. Counting the Multiples of 60

To find how many multiples of 60 exist in the range from 1 to 1000, we can divide the upper limit of the range by 60 and take the integer part of the result.

Number of multiples = floor(1000 / 60)

1000 ÷ 60 = 100 ÷ 6 = 16.66...

The integer part of this result is 16.

This means the multiples of 60 in the range are 60 × 1, 60 × 2, ..., up to 60 × 16 (which is 960).

Conclusion

There are 16 numbers between 1 and 1000 that are divisible by all the numbers 2, 3, 4, 5, and 6.